Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Friday, August 21, 2009

An application of VDW theorem to Number Theory- is there a better proof?

I present what may have been the first Application of van der Waerden's Theorem.
I also ask the question: Is there an alternative proof? This would be interesting since
the hope is that an alternative proof would have better bounds.

Notation: [W] is the set {1,...,W}, QR means Quadratic Residue (square root mod p,
where p will be understood), QNR means NOT a Quadratic Residue.

VDW Theorem: For all k, for all c, there exists W such that for all c-colorings
of [W] there exists a,d (d &ge 1) such that
a, a+d, ..., a+(k-1)d are all the same color.

One might wonder- can we also have d be that color?
How about a multiple of d?
OKAY, you might not wonder that, but the answer is YES and
we need this extension of VDW for our application:

Extension of VDW Theorem: For all k, for all s, for all c, there exists W such that for all c-colorings
of [W] there exists a,d (d &ge 1) such that
a, a+d, ..., a+(k-1)d, sd are all the same color.

See
this excerpt from my book
for a proof.
(We only need the s=1 case, but this version with general s is no harder and
is used in a proof of Rado's theorem.)

Before presenting the theorem duh jour and its proof we quote
Karen Johannson's excellent Masters Thesis
Variations on a theorem by van der Waerden
(2007 from The University of Manitoba, Dept of Mathematics)

Historically, the first application of van der Waerden's theorem may be due to
Brauer who proved a conjecture of Schur about quadratic residues. Brauer used
a generalization of van der Waerden's theorem, which he attributed to Schur.
The following theorem is a further
generalization of Brauer's result. The proof is now folklore and I have been unable
to locate the original source. The details appear, for example, in (she gives ref to
TO GRS book on RAMSEY THEORY, page 70).
(She then gives the statement and proof of what I called
above Extension of VDW. I suspect that Brauer only proved the s=1 case
since that is all he needed.)

We won't restate or prove Extension of VDW, but we give the theorem on
QR's that uses it.

Theorem:
For all k there exists p0 such that, for all primes p &gt p0
there are k consecutive QR in Zp (the integers mod p).

Proof: Let p0=W(2k+1,2).
Let p &gt p0.
Color [p] as follows:

COL(x) = 1 if x is a QR mod p, 0 otherwise.

By The Extended VDW theorem, with s=1, there is a, d such that

a, a+d, a+2d, ..., a+2kd, d

are either all QR or all QNR.
Let d-1 be the inverse of d mod p.
Since the product of two QR'is a a QR
and the product of two QNR's is a QR (that is not a typo- it really is true)
we have that

Recall that in VDW or extended VDW we getthat if we c-color[W] we get a monochromatick-AP. That entire k-AP iscontained in [W]. Hence wehave to have d &le W, and in particular we cannot havethat W divides d.

So when we 2-color [p]and use VDW the d that weget is such that d&le pand hence p cannot divide d.

My excuse: I was using the infinitary version of van der Waerden's theorem, and as such was coloring all of the integers. I don't know why I was doing that, considering your proof is very clear that you are using the version with bounds, and that you're coloring [p], not Z.

It feels like there should be some sort of argument making use of the fact that the Quadratic Residues are in a certain sense pseudorandom.

For example, if the appropriate Gowers norm of the set of residues is small, then we know that there should be approximately the same number of progressions in the residues as there are in a random set of density 1/2, which would be enough once p becomes much larger than 2^(2k).

One can look at the affine curve defined by x_1^2 + 1 = x_2^2 + 2 = x_3^2 + 3 = ... = x_k^2 + k modulo p, and then apply the Riemann hypothesis for curves (Weil) to it. If some things about irreducibility check out, then this should actually give an asymptotic on the number of solutions modulo p (about p/2^k?), not only their existence.

According to my colleague's calculations, the proof via the Riemann Hypothesis for curves (proven by Weil around 1940) gives a bound of k^2 * 4^(k+1) at worst, but this can likely be marginally improved.

Boris, if you or someone writes this up, please send it to me--- I am both interested in when Ramsey Theory is used to proof something AND when it is no longer needed.I would post a writeup of what I put on the blog AND your write up both on myWEBSITE OF RAMSEY APPLICATIONS.

Very cool application of VDW! I wonder if there is a compilation of surprising uses of VDW (or maybe Ramsey Theory) in atypical areas. I know a lot of topological proofs that use VDW but this is the first NT one I have seen. Again, very cool!